α-prime ideals

Let R be a commutative ring with identity. We give a new generalization to prime ideals called α-prime ideal. A proper ideal P of R is called an α-prime ideal if for all $a,b$ in R with ab∈ P, then either a∈ P or α(b)∈ P where α ∈ End(R). We study some properties of α-prime ideals analogous to prime ideals. We give some characterizations for such generalization and we prove that the intersection of all α-primes in a ring R is the set of all α-nilpotent elements in R. Finally, we give new versions of some famous theorems about prime ideals including α-integral domains and α-fields.